User tatin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:05:40Z http://mathoverflow.net/feeds/user/27832 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121615/subspace-of-skew-symmetric-matrices-of-rank-four Subspace of Skew-symmetric Matrices of Rank Four tatin 2013-02-12T15:54:02Z 2013-02-14T10:28:09Z <p>Let $n\geqslant 5$ and let $E_4(n)$ be a linear subspace of $(n\times n)$- real skew-symmetric matrices such that $$rank(A)=4,\text{ for all }A\in E_4(n),A\neq 0.$$ I'm curious about the following question:</p> <p>QUESTION:</p> <p>What can be said about the dimension of $E_4(n)$? Of course, it is easy to check that $\operatorname*{dim}E_4(n)\leqslant \binom{n-1}{2}$. But what is the best possible value? I'm specially interested in the case $n=5,6,7$.</p> http://mathoverflow.net/questions/113961/a-question-on-exterior-forms A Question on Exterior Forms tatin 2012-11-20T16:29:49Z 2013-02-12T16:34:17Z <p>For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions of a certain system of PDE. I have asked a special case of the problem in </p> <p>Link: <a href="http://mathoverflow.net/questions/112659/inequalities-involving-wedge-product-reference-request" rel="nofollow">http://mathoverflow.net/questions/112659/inequalities-involving-wedge-product-reference-request</a> </p> <p>with the hope that this particular case will do what I have in mind, but this did not turn out to be the case. Any help in this direction is welcome.</p> <p><em><strong>QUESTION:</em></strong> </p> <p>Let $N$ be a linear subspace of $\Lambda^2(\mathbb{R}^n)$ satisfying $$\omega\wedge\omega\neq 0,\text{ for all }\omega\in N,\omega\neq 0.$$ Is it true that there exists a $a\in\Lambda^4(\mathbb{R}^n)$, $a\neq 0$ such that $$\langle a;\omega\wedge\omega\rangle>0,\text{ for all }\omega\in N,\omega\neq 0?$$ <strong>Comment:</strong> My guess is that there will be such an $a$. But I could not prove it.</p> http://mathoverflow.net/questions/118037/on-the-positive-definiteness-of-a-linear-combination-of-matrices On the Positive Definiteness of a Linear Combination of Matrices tatin 2013-01-04T10:16:54Z 2013-01-07T09:43:42Z <p>Hello, </p> <p>In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.</p> <p><strong>QUESTION:</strong> </p> <p>Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, symmetric, indefinite matrices. I'm interested in conditions on $A_1,\ldots,A_m$ which ensures that the set <code> $$P:=\{\sum_{i=1}^{m}\lambda_i A_i:\lambda_i\in\mathbb{R}\}$$ </code>contains a positive-definite matrix. I'm aware of the following result due to Hestenes-McShane (1940) which is suffcient but not necessary.</p> <p><strong>THEOREM (Hestenes-McShane)</strong></p> <p>Let $m,n\in\mathbb{N}$ and let $A,B_i\in M_n(\mathbb{R})$ be real symmetric matrices, for all $i=1,\ldots,m$. Let us write, for each $i=1,\ldots,m$, <code> $$Z_{i}:=\{x\in\mathbb{R}^n:\langle B_i x;x \rangle=0\}.$$ </code> Let us suppose that</p> <ol> <li><p>$\langle A x;x \rangle>0$, for all $x\in \cap_{i=1}^{m}Z_i$, $x\neq 0$.</p></li> <li><p>$B$ is indefinite on $\mathbb{R}^n$, for all non-zero <code>$B\in\operatorname*{span}\{B_i:i=1,\ldots,m\}$</code>.</p></li> <li><p>For every non-zero subspace $S\subseteq \mathbb{R}^n$ satisfying $$S\cap\left(\cap_{i=1}^{m}Z_i\right)={0},$$ there exists <code>$B\in\operatorname*{span}\{B_i:i=1,\ldots,m\}$</code> such that $B$ is positive definite on $S$.</p></li> </ol> <p>Then, there exists <code>$B\in\operatorname*{span}\{B_i:i=1,\ldots,m\}$</code> such that $A-B$ is positive definite on $\mathbb{R}^n$.</p> <p>Unfortunately, in my case, condition 3 is not satisfied. Has this result been improved later?</p> http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts Classical Derivative, Weak Derivative and Integration by Parts tatin 2012-12-17T09:59:42Z 2012-12-17T21:18:55Z <p>Hello,</p> <p>While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated. </p> <p><strong>QUESTION</strong></p> <p>Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $f\in L^1_{\text{loc}}(U)$ such that </p> <p>1) the classical derivative $Df$ exists everywhere in $U$.</p> <p>2) $f$ is weakly differentiable in $U$. Let us write $D_w f$ to denote the weak derivative of $f$.</p> <p>3) $Df\neq D_w f$, on a set of <strong>positive measure</strong>.</p> <p><strong>Note that, we are assuming the existence of both the derivatives.</strong> I'm aware of examples where one exists while other one does not. </p> <p>The problem seems to be related to the question of validity of integration by parts for functions that are only differentiable. </p> <p>Thank you.</p> http://mathoverflow.net/questions/115427/pde-with-the-jacobian-determinant PDE with the Jacobian Determinant tatin 2012-12-04T17:33:48Z 2012-12-05T20:38:03Z <p>Hello,</p> <p>Could you please help me in answering the following question?</p> <p>Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not been able to control the support. </p> <p>The question seems so natural that I'm sure it has been studied somewhere in some form but I have not been able to dig out the right reference. Few minutes in MathScinet did not come up with what I'm looking for. All references seem to handle the situation when $f$ is positive which is clearly not the case here. Any suggestion or reference in this direction is welcome.</p> <p>Thank you.</p> <p>QUESTION: </p> <p>Let $U\subset\mathbb{R}^n$ be open, connected and let $f\in C_{0}^{\infty}(U;\mathbb{R})$ satisfy $$\int_{U}f(x)dx=0.$$ Is it true that there exists a $u\in C_{0}^{\infty}(U;\mathbb{R}^n)$ satisfying $$\operatorname*{det}(\nabla u)=f \text{ in }U?$$ Note that <code>$$C_{0}^{\infty}(U;\mathbb{R}^n)=\{u\in C^{\infty}(U;\mathbb{R}^n):\operatorname*{supp}(u)\text{ is compact and }\operatorname*{supp}(u)\subset U\}.$$</code></p> http://mathoverflow.net/questions/112659/inequalities-involving-wedge-product-reference-request Inequalities Involving Wedge Product (Reference Request) tatin 2012-11-17T08:52:18Z 2012-11-18T14:16:56Z <p>Hello, </p> <p>I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The article of Iwaniec-Kauhanen-Kravetz-Scott may be quoted as an example. </p> <p>More specifically, I'm interested in the validity of Cauchy-Schwarz type inequality with respect to the wedge product. As a very special case of what I have in mind, I'm interested the necessary and sufficient condition (or some reasonable sufficient condition) for the existence of $b\in\Lambda^4(\mathbb{R}^n),b\neq 0$ satisfying $$\langle b;\omega_1\wedge\omega_2\rangle^2&lt; \langle b;\omega_1^2\rangle \langle b;\omega_2^2\rangle,\ \langle b;\omega_1^2\rangle>0, \langle b;\omega_2^2\rangle>0,$$ where $\omega_1,\omega_2\in\Lambda^2(\mathbb{R}^n)$ is given.</p> <p>Thank you.</p> http://mathoverflow.net/questions/112360/decomposability-of-exterior-two-forms Decomposability of exterior two-forms tatin 2012-11-14T07:40:40Z 2012-11-14T14:25:43Z <p>Hello, </p> <p>The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is welcome. </p> <p>Thank you. </p> <p>Question:</p> <p>Let $u,v\in \Lambda^2$ be such that $$u\wedge u=-v\wedge v\neq 0.$$ Does there exist $\alpha,\beta\in\mathbb{R}$ such that $$(\alpha u+\beta v)\wedge (\alpha u+\beta v)=0\text{ and }\alpha u+\beta v\neq 0?$$</p> http://mathoverflow.net/questions/121615/subspace-of-skew-symmetric-matrices-of-rank-four/121619#121619 Comment by tatin tatin 2013-02-12T16:36:52Z 2013-02-12T16:36:52Z You are right. I'm referring to linear subspace. Let me modify the question so as to avoid the confusion. http://mathoverflow.net/questions/121615/subspace-of-skew-symmetric-matrices-of-rank-four/121619#121619 Comment by tatin tatin 2013-02-12T16:32:44Z 2013-02-12T16:32:44Z I do not understand one point. You mentioned that $E_4(5)=10$. Let us take $S:=e_1\wedge\mathbb{R}^5$. Then, S is a subspace of $5\times 5$ skew-symmetric matrices that has trivial intersection with $E_4(5)$. This implies that dimension of $E_4(5)$ has to be less than or equal to 10-4=6. http://mathoverflow.net/questions/120762/non-negative-quadratic-forms-with-exterior-forms Comment by tatin tatin 2013-02-04T14:09:00Z 2013-02-04T14:09:00Z It may have some some connection with the following thread: <a href="http://mathoverflow.net/questions/118037/on-the-positive-definiteness-of-a-linear-combination-of-matrices" rel="nofollow" title="on the positive definiteness of a linear combination of matrices">mathoverflow.net/questions/118037/&hellip;</a> http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts/116642#116642 Comment by tatin tatin 2012-12-18T02:47:51Z 2012-12-18T02:47:51Z Let me say it again: I'm sorry for the confusion. I have had the impression that the term &quot;weak derivative&quot; is used when the distributional derivative is a $L^1_{loc}$ function. I thought that this is a standard convention but, as it seems, I'm wrong. http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts Comment by tatin tatin 2012-12-18T02:46:35Z 2012-12-18T02:46:35Z I'm sorry for the confusion. I have had the impression that the term &quot;weak derivative&quot; is used when the distributional derivative is a $L^1_{loc}$ function. I thought that this is a standard convention but, as it seems, I'm wrong. http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts Comment by tatin tatin 2012-12-17T18:42:21Z 2012-12-17T18:42:21Z The way it is defined in the context of Sobolev spaces http://mathoverflow.net/questions/116580/classical-derivative-weak-derivative-and-integration-by-parts/116585#116585 Comment by tatin tatin 2012-12-17T12:08:38Z 2012-12-17T12:08:38Z That is precisely the question. Is there any integration by parts if one function is merely differentiable? http://mathoverflow.net/questions/115427/pde-with-the-jacobian-determinant Comment by tatin tatin 2012-12-05T17:19:57Z 2012-12-05T17:19:57Z @Robert Bryant: Thank you. Connectedness is an assumption here. Re-edited accordingly. http://mathoverflow.net/questions/115427/pde-with-the-jacobian-determinant Comment by tatin tatin 2012-12-05T05:42:41Z 2012-12-05T05:42:41Z @Ryan budney: What you are talking about is the problem with prescribed divergence which is essentially the linearized version of the problem with prescribed Jacobian. http://mathoverflow.net/questions/113961/a-question-on-exterior-forms/114116#114116 Comment by tatin tatin 2012-11-22T06:59:50Z 2012-11-22T06:59:50Z @Sergei: Thank you. Very nice example!!. http://mathoverflow.net/questions/113961/a-question-on-exterior-forms/113990#113990 Comment by tatin tatin 2012-11-21T18:02:22Z 2012-11-21T18:02:22Z Thanks a lot! I do not know how to thank you enough!!! http://mathoverflow.net/questions/113961/a-question-on-exterior-forms/113990#113990 Comment by tatin tatin 2012-11-21T04:09:25Z 2012-11-21T04:09:25Z Thanks again. Thank a lot. I'm not familiar with representation theory. It will take me some time to understand your counterexample fully. Is there any direct counterexample that constructs the subspace explicitly? http://mathoverflow.net/questions/113961/a-question-on-exterior-forms Comment by tatin tatin 2012-11-20T18:52:01Z 2012-11-20T18:52:01Z I have modified so to avoid any confusion. http://mathoverflow.net/questions/113961/a-question-on-exterior-forms Comment by tatin tatin 2012-11-20T18:50:26Z 2012-11-20T18:50:26Z Yes. A vector subspace. http://mathoverflow.net/questions/113961/a-question-on-exterior-forms Comment by tatin tatin 2012-11-20T18:29:57Z 2012-11-20T18:29:57Z Note that, your example is not the right one. For $\omega_{+}$,$\omega_{-}$ both cannot be in $N$ as $\omega_{+}+\omega_{-}=2dx_1\wedge dy_1$ which is decomposable whereas $N$ cannot contain non-zero decomposable forms.